\[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{5 e \left (e x +d \right )^{\frac {5}{2}}}+\frac {2 \left (b^{2} e^{2}-16 b c d e +16 c^{2} d^{2}\right ) \EllipticE \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {c}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {e x +d}}{5 d \,e^{4} \left (-b e +c d \right ) \sqrt {1+\frac {e x}{d}}\, \sqrt {c \,x^{2}+b x}}-\frac {16 \left (-b e +2 c d \right ) \EllipticF \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {c}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {1+\frac {e x}{d}}}{5 e^{4} \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}-\frac {2 \left (c \,d^{2} \left (-7 b e +8 c d \right )+e \left (b^{2} e^{2}-10 b c d e +10 c^{2} d^{2}\right ) x \right ) \sqrt {c \,x^{2}+b x}}{5 d \,e^{3} \left (-b e +c d \right ) \left (e x +d \right )^{\frac {3}{2}}} \]
command
integrate((c*x^2+b*x)^(3/2)/(e*x+d)^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left ({\left (16 \, c^{3} d^{6} + b^{3} x^{3} e^{6} + 3 \, {\left (2 \, b^{2} c d x^{3} + b^{3} d x^{2}\right )} e^{5} - 3 \, {\left (8 \, b c^{2} d^{2} x^{3} - 6 \, b^{2} c d^{2} x^{2} - b^{3} d^{2} x\right )} e^{4} + {\left (16 \, c^{3} d^{3} x^{3} - 72 \, b c^{2} d^{3} x^{2} + 18 \, b^{2} c d^{3} x + b^{3} d^{3}\right )} e^{3} + 6 \, {\left (8 \, c^{3} d^{4} x^{2} - 12 \, b c^{2} d^{4} x + b^{2} c d^{4}\right )} e^{2} + 24 \, {\left (2 \, c^{3} d^{5} x - b c^{2} d^{5}\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right ) + 3 \, {\left (16 \, c^{3} d^{5} e + b^{2} c x^{3} e^{6} - {\left (16 \, b c^{2} d x^{3} - 3 \, b^{2} c d x^{2}\right )} e^{5} + {\left (16 \, c^{3} d^{2} x^{3} - 48 \, b c^{2} d^{2} x^{2} + 3 \, b^{2} c d^{2} x\right )} e^{4} + {\left (48 \, c^{3} d^{3} x^{2} - 48 \, b c^{2} d^{3} x + b^{2} c d^{3}\right )} e^{3} + 16 \, {\left (3 \, c^{3} d^{4} x - b c^{2} d^{4}\right )} e^{2}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right )\right ) + 3 \, {\left (8 \, c^{3} d^{4} e^{2} - 11 \, b c^{2} d x^{2} e^{5} + b^{2} c x^{2} e^{6} + {\left (11 \, c^{3} d^{2} x^{2} - 16 \, b c^{2} d^{2} x\right )} e^{4} + {\left (18 \, c^{3} d^{3} x - 7 \, b c^{2} d^{3}\right )} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x e + d}\right )}}{15 \, {\left (c^{2} d^{5} e^{5} - b c d x^{3} e^{9} + {\left (c^{2} d^{2} x^{3} - 3 \, b c d^{2} x^{2}\right )} e^{8} + 3 \, {\left (c^{2} d^{3} x^{2} - b c d^{3} x\right )} e^{7} + {\left (3 \, c^{2} d^{4} x - b c d^{4}\right )} e^{6}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} \sqrt {e x + d}}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}, x\right ) \]